SFGate

yvesun Most split levels such as yours don't have that door. Remove the door and hinges, it looks like it was added at some point after the house was built. Just remember to put up a hand rail.

Adrian Ramsay Design House Hey Kelly, like your idea
before doing to much call a builder and ask some questions,
I would imagine there is a support post to the left of the door that creates the corner of the stairs and above/checked into this there is a beam that runs to the wall on the right.
You should be safe to remove all the wall boards and open the space up so you can see what it will look like as we'll as checking what's structural and what's not. It's only when you start to remove any upright posts that you could have problems.

Cheers Adrian
ARDH

AFC inc Depending from age of your home , area where it located the stairs could be done in different ways . Inspect carefully stairs conditions before you make any demolitions.

Margaret Phillips My parent built a very similar house in 1958 and my Mother still lives there. They actually added THAT door when I was little (I was the youngest of five) just to be able to cut the noise down of five kids playing and TV noise. My Mother removed the door in the last twenty years sometime. I would love to see your project of opening up the entire stairwell. It would really awesome. In my parent's home there I are several steel columns that go from the basement to I guess supporting the uppermost level, three levels of floors. Getting an engineer out to inspect your house likely would only cost a few hundred dollars, a few hundred more if you want a signed and stamped official statement. I had one out to my house when about six of us experienced DIY renovators could not agree on what walls could be taken out, even with two engineers among us. This will go a long way in peace of mind when ripping out walls. If your house is like my parent's house it is likely you can take those walls out to have a multi-level stairwell.

AFC inc It looks like this can be done without any problems . As soon as it is not a bearing walls ( which is relatively easy to find ) should not be a big deal.
The open stairs should have return at the ends and cove molding . Which is not hard to do and looks much better . Unfortunately old steps most possible can not be saved because new one should be longer. I dont know where a you live but we live on West Coast Portland Oregon.
Although we do sometimes cabinets work in other states ( South Florida , South California , Hawaii etc ) there is not sense to travel a very long distance .
In my opinion this could be a few days of work and half of the time will be spend on demolitions which does not require a lot of skills. The skilled carpentry work may take a few days .
As for engineer you may be right . Please feel free to ask any questions .
Here is the picture of stairway we did last year . One is free standing and did not touched the walls.

AFC inc Free standing curved stairway . Materials Claro walnut